Results 1  10
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28
Sudden Emergence Of A Giant kCore In A Random Graph.
 J. Combinatorial Theory, Series B
, 1996
"... The k core of a graph is the largest subgraph with minimum degree at least k . For the ErdosR'enyi random graph G(n; m) on n vertices, with m edges, it is known that a giant 2core grows simultaneously with a giant component, that is when m is close to n=2 . We show that for k 3 , with high proba ..."
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Cited by 104 (8 self)
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The k core of a graph is the largest subgraph with minimum degree at least k . For the ErdosR'enyi random graph G(n; m) on n vertices, with m edges, it is known that a giant 2core grows simultaneously with a giant component, that is when m is close to n=2 . We show that for k 3 , with high probability, a giant k core appears suddenly when m reaches c k n=2 ; here c k = min ?0 = k () and k () = PfPoisson() k \Gamma 1g . In particular, c 3 3:35 . We also demonstrate that, unlike the 2core, when a k core appears for the first time it is very likely to be giant, of size p k ( k )n . Here k is the minimum point of = k () and p k ( k ) = PfPoisson( k ) kg . For k = 3 , for instance, the newborn 3core contains about 0:27n vertices. Our proofs are based on the probabilistic analysis of an edge deletion algorithm that always finds a k core if the graph has one. 1991 Mathematics Subject Classification. Primary 05C80, 05C85, 60C05; Secondary 60F10, 60G42, 60J10.
Random maps, coalescing saddles, singularity analysis, and Airy phenomena
 Random Structures & Algorithms
, 2001
"... A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponentialcubic type, corresponding to distributions that involve the ..."
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Cited by 45 (6 self)
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A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponentialcubic type, corresponding to distributions that involve the Airy function. In this paper, such Airy phenomena are related to the coalescence of saddle points and the confluence of singularities of generating functions. For about a dozen types of random planar maps, a common Airy distribution (equivalently, a stable law of exponent 3/2) describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and fine optimization of random generation algorithms for multiply connected planar graphs. Based on an extension of the singularity analysis framework suggested by the Airy case, the paper also presents a general classification of compositional schemas in analytic combinatorics.
Percolation on Transitive Graphs as a Coalescent Process: Relentless Merging Followed by Simultaneous Uniqueness
 IN PERPLEXING PROBABILITY PROBLEMS: PAPERS IN HONOR OF
, 1998
"... Consider i.i.d. percolation with retention parameter p on an infinite graph G. There is a well known critical parameter p c 2 [0; 1] for the existence of infinite open clusters. Recently, it has been shown that when G is quasitransitive, there is another critical value p u 2 [p c ; 1] such that th ..."
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Cited by 27 (9 self)
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Consider i.i.d. percolation with retention parameter p on an infinite graph G. There is a well known critical parameter p c 2 [0; 1] for the existence of infinite open clusters. Recently, it has been shown that when G is quasitransitive, there is another critical value p u 2 [p c ; 1] such that the number of infinite clusters is a.s. 1 for p 2 (p c ; p u ), and there is a unique infinite cluster for p ? p u . We prove a simultaneous version of this result in the canonical coupling of the percolation processes for all p 2 [0; 1]; in particular, a.s. simultaneous uniqueness holds for all p ? p u . Simultaneously for all p 2 (p c ; p u ), we also prove that each infinite cluster has uncountably many ends. For p ? p c we prove that all infinite clusters are indistinguishable by robust properties. Under the additional assumption that G is unimodular, we prove that a.s. for all p 1 ! p 2 in (p c ; p u ), every infinite cluster at level p 2 contains infinitely many infinite clusters at lev...
Analytic Variations On The Airy Distribution
, 2001
"... . The Airy distribution (of the \area" type) occurs as limit distribution of cumulative parameters in a number of combinatorial structures, like path length in trees, area below walks, displacement in parking sequences, and it is also related to basic graph and polyomino enumeration. We obtain curio ..."
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Cited by 23 (4 self)
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. The Airy distribution (of the \area" type) occurs as limit distribution of cumulative parameters in a number of combinatorial structures, like path length in trees, area below walks, displacement in parking sequences, and it is also related to basic graph and polyomino enumeration. We obtain curious explicit evaluations for certain moments of the Airy distribution, including moments of orders 1; 3; 5; &c, as well as + 1 3 ; 5 3 ; 11 3 ; &c. and 7 3 ; 13 3 ; 19 3 ; &c . Our proofs are based on integral transforms of the Laplace and Mellin type and they rely essentially on \nonprobabilistic" arguments like analytic continuation. A byproduct of this approach is the existence of relations between moments of the Airy distribution, the asymptotic expansion of the Airy function Ai(z) at +1, and power symmetric functions of the zeros k of Ai(z). For probabilists, the Airy distribution considered here is nothing but the distribution of the area under the Brownian excursion. The ...
Phase transition and finitesize scaling for the integer partitioning problem
, 2001
"... Dedicated to D. E. Knuth on the occasion of his 64th birthday. Abstract. We consider the problem of partitioning n randomly chosen integers between 1 and 2 m into two subsets such that the discrepancy, the absolute value of the difference of their sums, is minimized. A partition is called perfect if ..."
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Cited by 17 (2 self)
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Dedicated to D. E. Knuth on the occasion of his 64th birthday. Abstract. We consider the problem of partitioning n randomly chosen integers between 1 and 2 m into two subsets such that the discrepancy, the absolute value of the difference of their sums, is minimized. A partition is called perfect if the optimum discrepancy is 0 when the sum of all n integers in the original set is even, or 1 when the sum is odd. Parameterizing the random problem in terms of κ = m/n, we prove that the problem has a phase transition at κ = 1, in the sense that for κ < 1, there are many perfect partitions with probability tending to 1 as n → ∞, while for κ> 1, there are no perfect partitions with probability tending to 1. Moreover, we show that this transition is firstorder in the sense the derivative of the socalled entropy is discontinuous at κ = 1. We also determine the finitesize scaling window about the transition point: κn = 1 − (2n) −1 log 2 n + λn/n, by showing that the probability of a perfect partition tends to 1, 0, or some explicitly computable p(λ) ∈ (0, 1), depending on whether λn tends to −∞, ∞, or λ ∈ (−∞, ∞), respectively. For λn → − ∞ fast enough, we show that the number of perfect partitions is Gaussian in the limit. For λn → ∞, we prove that with high probability the optimum partition is unique, and that the optimum discrepancy is Θ(2 λn). Within the window, i.e., if λn  is bounded, we prove that the optimum discrepancy is bounded. Both for λn → ∞ and within the window, we find the limiting distribution of the (scaled) discrepancy. Finally, both for the integer partitioning problem and for the continuous partitioning problem, we find the joint distribution of the k smallest discrepancies above the scaling window.
Some Large Deviation Results for Sparse Random Graphs
, 1998
"... We obtain a large deviation principle (LDP) for the relative size of the largest connected component in a random graph with small edge probability. The rate function, which is not convex in general, is determined explicitly using a new technique. The proof yields an asymptotic formula for the probab ..."
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Cited by 16 (0 self)
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We obtain a large deviation principle (LDP) for the relative size of the largest connected component in a random graph with small edge probability. The rate function, which is not convex in general, is determined explicitly using a new technique. The proof yields an asymptotic formula for the probability that the random graph is connected. We also present an LDP and related result for the number of isolated vertices. Here we make use of a simple but apparently unknown characterisation, which is obtained by embedding the random graph in a random directed graph. The results demonstrate that, at this scaling, the properties `connected' and `contains no isolated vertices' are not asymptotically equivalent. (At the threshold probability they are asymptotically equivalent.) Keywords: giant component, connectivity, isolated vertices. 1 1 Introduction The central object of study in this paper is the random graph G(n; p), with p = O(1=n). The random graph G(n; p) is constructed on n vertice...
A phase transition in the random transposition random walk
 Pages 1726 in Banderier and Krattenthaler (2003) Bollobás, B
, 2003
"... Our work is motivated by Bourque and Pevzner’s (2002) simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk on the group of permutations on n elements. Consider this walk in continuous time ..."
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Cited by 12 (7 self)
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Our work is motivated by Bourque and Pevzner’s (2002) simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk on the group of permutations on n elements. Consider this walk in continuous time starting at the identity and let Dt be the minimum number of transpositions needed to go back to the identity from the location at time t. Dt undergoes a phase transition: the distance D cn/2 ∼ u(c)n, where u is an explicit function satisfying u(c) =c/2 for c ≤ 1 and u(c) <c/2 for c>1. In addition, we describe the fluctuations of D cn/2 about its mean in each of the three regimes (subcritical, critical and supercritical). The techniques used involve viewing the cycles in the random permutation as a coagulationfragmentation process and relating the behavior to the ErdősRenyi random graph model.
Planar Maps and Airy Phenomena
, 2000
"... A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type (e x 2 ), that is, Gaussian. We exhibit here a new class of \universal" phenomena that are of the exponentialcubic type (e ix 3 ), corresponding to ..."
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Cited by 12 (4 self)
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A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type (e x 2 ), that is, Gaussian. We exhibit here a new class of \universal" phenomena that are of the exponentialcubic type (e ix 3 ), corresponding to nonstandard distributions that involve the Airy function. Such Airy phenomena are expected to be found in a number of applications, when conuences of critical points and singularities occur. About a dozen classes of planar maps are treated in this way, leading to the occurrence of a common Airy distribution that describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and ne optimization of random generation algorithms for multiply connected planar graphs.
Asymptotic distributions for the cost of linear probing hashing, Random Structures and Algorithms
"... Abstract. We study moments and asymptotic distributions of the construction cost, measured as the total displacement, for hash tables using linear probing. Four different methods are employed for different ranges of the parameters; together they yield a complete description. This extends earlier res ..."
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Cited by 11 (3 self)
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Abstract. We study moments and asymptotic distributions of the construction cost, measured as the total displacement, for hash tables using linear probing. Four different methods are employed for different ranges of the parameters; together they yield a complete description. This extends earlier results by Flajolet, Poblete and Viola. The average cost of unsuccessful searches is considered too. 1.